I would go lei shi>vizier>protectors(havent downed them yet but all the others)>ambershaper>empress

vizier and ambershaper were a pain in the ass for our raid group, empress was pretty easy with 2 discipline priest(your shadow should get some heal gear :P ) lei shi died after like 7 pulls.
Lei Shi is by far the easiest, just handle the scary fog right and you should be fine. and it is dropping a T token

difficulty wise it depends. vizier is easiest if your group is good on awareness(disks), ambershaper is easy if at least 1 of ur tanks are good and can handle being in the construct the entire phase 2 keeping stacks on the boss + interrupting self while anyone else who gets MC'd interrupts themselves and the monstrosity before being broken out. lei shi and protectors are dps/heals heavy with lei shi being rng based(getting more hides than get aways) and beating the enrage.

Vizier can be very RNG dependant with the MCs and attenuation in P3 but shouldn't be too difficult.

Amber shaper is just a test for the retards in your guild with interupt yourself/the boss however not terribly difficult if you have a tank who knows what to do. Personally I'd try and get those two down as Empress is surprisingly easy. Following that try protectors or lei shi, different groups will do well at different bosses, not too sure how your aoe will go on lei shi to break the hides but

Originally Posted by Zarhym
(Blue Tracker)

We're confident the .01% left will be thrilled to attend a post-apocalyptic BlizzCon.

Having only killed Lei Shi of the above, but tried both Amber Shaper and Zorlok, I would advise you to go for Lei Shi if your DPS are good.
It might be a bit tricky for your comp to handle adds, though. But you have mass fear, which is really good for the add phases. It might be necessary for your shaman to spec into Frozen Power to perma root an add which makes 5 reliable CCs and Blind for the 20% phase.

Based on that comp, what would the recommended order be? Thanks in advance, it's one thing to talk about it amongst ourselves but feedback from those ahead of us currently would be great to hear.

Hey, it really depends on where the stronger players in your group are.

Vizier and Amber Shaper place a pretty high requirement on every individual in your raid, so if you don't do well on individual responsibility fights (have a couple people who let down the group) then you'll struggle a bit more on those two. If your players are excellent but lack gear I would suggest do these fights because the dps and healing requirements are not that significant but the awareness/player requirements are higher.

If your range dps are strong players then Protectors is good for you as managing the stacks from the adds is the most important part of the fight. Also healers have to be pretty good if you opt to 2heal, as the Touch of Sha debuff damage becomes quite big and then the longer they can keep you up for the soft enrage, the easier it is to kill. Interrupts are pretty important too but not all that complex if your melee pay attention. Plus you have a prot paladin who has extra interrupts with shield.

Lei Shi is probably the easiest of the four you mentioned, just have to beat the enrage, but again you need range dps to be able to manage the stacks (not too challenging). The only other difficult part of this fight on 10m is CCing the Elementals as they spawn.

No, you got my question right away but Hide is only a minor thing in that fight, we have more trouble with dealing with the adds

Yes absolutely, the adds will fuck your tanks up, or your healers if your CC isnt quick.
Quick and coordinated CC is the best way to streamline the fight.
Don't forget that adds spawn every 20% on the dot (80, 60, 40, 20) so have your CC'ers (shaman, hunter, priest i guess) spread out and ready to CC the closest guy to them.